Alex Jones is being sued over Infowars’ spread of false claims that the 2012 shooting at Sandy Hook Elementary School in Newtown, Conn., was an elaborate hoax.CreditCreditTamir Kalifa/Austin American-Statesman, via Associated Press

AUSTIN, Tex. — In the five years since Noah Pozner was killed at Sandy Hook Elementary School in Newtown, Conn., death threats and online harassment have forced his parents, Veronique De La Rosa and Leonard Pozner, to relocate seven times. They now live in a high-security community hundreds of miles from where their 6-year-old is buried.

“I would love to go see my son’s grave and I don’t get to do that, but we made the right decision,” Ms. De La Rosa said in a recent interview. Each time they have moved, online fabulists stalking the family have published their whereabouts.

“With the speed of light,” she said. “They have their own community, and they have the ear of some very powerful people.”

On Wednesday in an Austin courtroom, the struggle of the Sandy Hook families to hold to account Alex Jones, a powerful leader of this online community, will reach a crossroads. Lawyers for Noah Pozner’s parents will seek to convince a Texas judge that they — and by extension the families of eight other victims in the 2012 shooting that killed 20 first graders and six adults — have a valid defamation claim against Mr. Jones, whose Austin-based Infowars media operation spread false claims that the shooting was an elaborate hoax.

The Pozner hearing is a bellwether in three cases, including another in Texas and one in Connecticut, filed by relatives of nine Sandy Hook victims. It comes as the social media platforms Mr. Jones relies upon to spread incendiary claims initiate efforts to curb him.

The day after the Pozner case, in the same courthouse, is a hearing in a separate defamation case against Mr. Jones brought by Marcel Fontaine, who was falsely identified on Infowars’ website as the gunman in the Parkland, Fla., school shooting in February. Mr. Fontaine, who lives in Massachusetts, has never visited Florida. The Pozner family and Mr. Fontaine are being represented by Mark Bankston of Farrar & Ball, a law firm based in Houston.

Mr. Jones is trying to have the Pozner and Fontaine cases dismissed under the Texas Citizens Participation Act, which protects citizens’ right to free speech against plaintiffs who aim to silence them through costly litigation. Mr. Jones is seeking more than $100,000 in court costs from the Pozner family. Efforts to reach Mr. Jones on his cellphone and through the Infowars email were unsuccessful. Mark Enoch, his lawyer in the case, did not respond to telephone and email requests for comment.

Mr. Jones has emerged as an avatar for a “post truth” ethos that flourished online during the last presidential campaign. He gained a national spotlight and millions of followers after Donald J. Trump appeared on his show during the campaign, praising his reputation as “amazing.” Since then, many of Mr. Jones’s bogus theories have targeted President Trump’s perceived adversaries and reflect opinions held by his political base.

Last week Mr. Jones broadcast a bizarre accusation that Robert S. Mueller III, the special counsel investigating Russian interference in the 2016 election, was involved in a child sex ring. In an online broadcast, Mr. Jones addressed Mr. Mueller while repeatedly imitating firing a handgun, saying: “It’s the real world. Politically. You’re going to get it, or I’m going to die trying.”

On Friday, Facebook removed Mr. Jones’s personal page from the site for 30 days, citing “bullying” and “hate speech.” The company has yet to take any action against the Infowars Facebook page, despite pleas from Sandy Hook parents. Last week, YouTube removed four of Mr. Jones’s videos for violating its child endangerment and hate speech standards and barred him from live-streaming on the site for 90 days, though his channel remains a main source for his broadcasts.

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Veronique De La Rosa, whose 6-year-old son, Noah Pozner, was killed in the shooting, has had to relocate several times because of online harassment and death threats.CreditJessica Hill/Associated Press

The videos removed included one called “Prevent Liberalism,” in which a man chokes a child and throws him to the ground. Another one was called “Shock Report: Learn How Islam Has Already Conquered Europe.”

Mr. Jones and his lawyers say in court filings that the Pozner family’s suit is an effort “to silence those who openly oppose their very public ‘herculean’ efforts to ban the sale of certain weapons, ammunition and accessories, to pass new laws relating to gun registration and to limit free speech.”

That stance echoes Mr. Jones’s original false claims that Sandy Hook was staged by government-backed gun control activists. Mr. Jones’s lawyer says in court filings that Mr. Jones’s theories are opinion, which is more broadly protected by the First Amendment. He also maintains that Mr. Pozner and Ms. De La Rosa are public figures, because Mr. Pozner has created a nonprofit to combat the harm caused by online falsehoods and Ms. De La Rosa has advocated a ban on assault weapons, like the AR-15-style rifle used in the Sandy Hook shooting.

If the parents are found to be public figures, they will have to prove actual malice, or that Mr. Jones knew the claims were false, but repeated them anyway.

The Pozner family’s story is recounted in the court filings: In 2015, after Mr. Pozner succeeded in having an Infowars video taken down from Mr. Jones’s YouTube channel, “Mr. Jones went on an angry rant about me for nearly an hour,” Mr. Pozner said in an affidavit. Mr. Jones “also hosted a call with an obsessed fellow conspiracy theorist who issued a threat to me.”

“Mr. Jones then showed his audience my personal information and maps to addresses associated with my family,” the affidavit says.

Lucy Richards, an Infowars devotee, was arrested the next year for repeatedly threatening Mr. Pozner’s life. She was sentenced to five months in prison last year. As a condition of parole, a judge ordered that she cease consuming Infowars programming, the court documents state.

“This type of misinformation is a bit of a societal crisis,” Ms. De La Rosa said. “This isn’t someone on a soapbox in Times Square spewing nonsense. It’s someone who every day generates income from his demonstrably false utterances.”

Starting days after the shooting, Mr. Jones helped spread false claims that the Sandy Hook parents were “crisis actors” in a government conspiracy. A month after the shooting, he began broadcasting excerpts from Ms. De La Rosa’s interview with the CNN anchor Anderson Cooper, which was taped in front of the Edmond Town Hall in Newtown. Mr. Jones falsely claimed that the interview was taped in a studio before a “green screen.”

In an April 2017 Infowars broadcast titled “Sandy Hook Vampires Exposed,” according to the court documents, Mr. Jones aired the clip of Ms. De La Rosa’s interview, noting that when Mr. Cooper turns his head, “his nose disappears repeatedly because the green screen isn’t set right.” The Pozner filings include an affidavit from Grant Fredericks, a forensic video analyst and expert witness, in which he attributes the anomaly in the videotape to “postproduction compression,” common in video production, and said “no credible video professional, editor or web-content specialist would conclude” that the interview was taped in front of a screen.

In court filings responding to the Pozner suit, Mr. Jones’s lawyer says that “certain comments that he had previously made in connection with possible faking or staging events at the Sandy Hook shooting was not an accusation against the children or families, but against the media and government officials, whose actions led many to question the official version of the event.” The parents, the filing says, “isolate specific statements, take them out of context while ignoring other relevant portions, and misinterpret what was said.”

Ms. De La Rosa and Mr. Pozner will not appear in court on Wednesday, in part because of safety concerns.

A version of this article appears in print on , on Page A13 of the New York edition with the headline: Judge to Rule on Sandy Hook Family’s Suit Against Infowars Founder. Order Reprints | Today’s Paper | Subscribe